Table of Contents

• • 0.1  First Order Shift, Constant Phase Shift
    • 0.1.1  Phase Ramp on Alternate Lines
    • 0.1.2  Fourier Transforms of the Phase Ramps
  • 0.2  Multiplication of K-space and Phase Ramps
  • 0.3  Convoluting Phase Rampss in Image Space
• 1  Inspecting the Blurring Caused by Alternate Line Phase Ramps
  • • 1.0.1  Seprate K-Space Lines that were shifted by Ramps
  • 1.1  Displaying the Individual Images and their Overlay
  • 1.2  Padding Lines of K-space to increase FOV

First Order Shift, Constant Phase Shift

import BrukerMRI as bruker
import pylab as pl
import plot
#Python method of extracting raw Bruker Files
import BrukerMRI as bruker
from jupyterthemes import jtplot
# currently installed theme will be used to
# set plot style if no arguments provided
jtplot.style(theme='onedork')
jtplot.style(ticks=True, grid=False, figsize=(6, 4.5))
# Modules
from mpl_toolkits.axes_grid1 import ImageGrid
import ipywidgets as widgets
from ipywidgets import interact, interactive_output, HBox
from mpl_toolkits.axes_grid1 import make_axes_locatable


import pandas as pd
import pydicom as dcm
import math
import numpy as np
import matplotlib.pyplot as plt
import glob
from os.path import expanduser
import os
from matplotlib import cm
import cmath

# copy and paste entire directory but replace "\" with "/"
# Place a "/" at the end of the line
MainDir = "C:/Users/daniala/Desktop/mri/20191210_104049_MRMD_phantom_practice_1_3/"
ExpNum = 8
Experiment = bruker.ReadExperiment(MainDir, ExpNum)
acqp = Experiment.acqp
method = Experiment.method
raw_fid = Experiment.raw_fid
k_space_gen = Experiment.GenerateKspace()
k_space = Experiment.k_data
k_sp = bruker.phase_shift(120,0,k_space)[:,:,0]

def k_plots(k_spaces):
    
    key_list = list(k_spaces)
    Nr = len(list(k_spaces))
    Nc = 2

    fig = plt.figure(figsize = (30,20))
    grid = ImageGrid(fig, 111,  # similar to subplot(111)
                     nrows_ncols=(Nr, Nc),  # creates 2x3 grid of axes
                     axes_pad=0.8,  # pad between axes in inch.
                     share_all=False,
                     cbar_location="right",
                     cbar_mode="each",
                     cbar_size="7%",
                     cbar_pad="5%",
                     )
    
    for key,value in k_spaces.items():
        c = key_list.index(key)
        
        im = grid[2*c].imshow(np.log(1+np.abs(value[:,:])))
        grid[2*c].set_xticks([])
        grid[2*c].set_yticks([])
        grid[2*c].set_title('k-space of ' + key)
        cb = grid.cbar_axes[2*c].colorbar(im)

        im2 = grid[1+2*c].imshow(abs(plot.fft_image(value[:,:])))
        grid[1+2*c].set_xticks([])
        grid[1+2*c].set_yticks([])
        grid[1+2*c].set_title('Image from ' + key)
        cb = grid.cbar_axes[1+2*c].colorbar(im2)

Phase Ramp on Alternate Lines

def phase_ramp_1(del_x,del_y,x,y,k_sp):
    shape = k_sp.shape
    N_x = shape[0]
    N_y = shape[1]
    
    p_ramp = np.ones((N_x,N_y),dtype='complex')
    ky = (np.linspace(0,N_y-1,N_y)-(N_y/2))/N_y  
    kx = (np.linspace(0,N_x-1,N_x)-(N_x/2))/N_x
    if x > 0:
        for i in range(0,N_x-1,x):
            p_ramp[i,:] = np.exp(-1j*del_x*kx*2*np.pi)
    if y > 0:         
       for i in range(0,N_y-1,y):
           p_ramp[:,i] = p_ramp[:,i] * np.exp(-1j*del_y*ky*2*np.pi)
        
    return(p_ramp)

first_x = phase_ramp_1(5,0,2,0,k_sp)
first_y = phase_ramp_1(0,5,0,2,k_sp)
first_xy = phase_ramp_1(5,5,2,2,k_sp)

Fourier Transforms of the Phase Ramps

first_x1 = phase_ramp_1(5,0,2,0,k_sp[0:30,0:30])
first_y1 = phase_ramp_1(0,5,0,2,k_sp[0:30,0:30])
first_xy1 = phase_ramp_1(5,5,2,2,k_sp[0:30,0:30])

k_spaces = {#'Original k-space':k_sp,
            'Phase Shift X':first_x1,
            'Phase Shift Y': first_y1,
            'Phase Shift XY': first_xy1
            }

k_plots(k_spaces)

C:\Users\daniala\AppData\Local\Continuum\anaconda3\lib\site-packages\ipykernel_launcher.py:25: UserWarning: Attempting to set identical bottom == top == 0.6931471805599453 results in singular transformations; automatically expanding.

Multiplication of K-space and Phase Ramps

k_first_x = np.multiply(k_sp,first_x)
k_first_y = np.multiply(k_sp,first_y)
k_first_xy = np.multiply(k_sp,first_xy)

k_spaces2 = {'Original k-space':k_sp,
            'Original with Ramp X': k_first_x,
            'Original with Ramp Y': k_first_y,
            'Original with Ramp XY': k_first_xy
            }
k_plots(k_spaces2)

Convoluting Phase Rampss in Image Space

from scipy import signal
image_first_x = signal.convolve2d(plot.fft_image(k_sp),plot.fft_image(first_x), boundary='symm', mode='same')
image_first_y = signal.convolve2d(plot.fft_image(k_sp),plot.fft_image(first_y), boundary='symm', mode='same')
image_first_xy = signal.convolve2d(plot.fft_image(k_sp),plot.fft_image(first_xy), boundary='symm', mode='same')

k_spaces1 = {'Original k-space':plot.fft_image(k_sp),
            'Convolution of Original Image with Phase Ramp X': image_first_x,
            'Convolution of Original Image with Phase Ramp Y': image_first_y,
            'Convolution of Original Image with Phase Ramp XY': image_first_xy,

            }

plot.comparison_i_plots(k_spaces1)

Inspecting the Blurring Caused by Alternate Line Phase Ramps

Seprate K-Space Lines that were shifted by Ramps

original = range(0,256,2)
shifted = range(1,256,2)

k_first_x_original = k_first_x[original]
k_first_x_shifted = k_first_x[shifted]
k_first_y_original = k_first_y[:,original]
k_first_y_shifted = k_first_y[:,shifted]

img_1 = np.fft.fftshift(np.fft.fft2(k_first_x_original))
img_2 = np.fft.fftshift(np.fft.fft2(k_first_x_shifted))

img_3 = np.fft.fftshift(np.fft.fft2(k_first_y_original))
img_4 = np.fft.fftshift(np.fft.fft2(k_first_y_shifted)) 

Displaying the Individual Images and their Overlay

plt.figure(figsize=(15,10))


plt.subplot(321)
plt.imshow(abs(img_1),cmap='PiYG')

plt.subplot(323)
plt.imshow(abs(img_2),cmap ='PiYG')

plt.subplot(325)
plt.imshow(abs(img_2),cmap ='PiYG')
plt.imshow(abs(img_1),cmap ='PiYG',alpha = 0.7)

plt.subplot(322)
plt.imshow(abs(img_3),cmap='plasma')

plt.subplot(324)
plt.imshow(abs(img_4),cmap ='plasma')

plt.subplot(326)
plt.imshow(abs(img_4),cmap ='plasma')
plt.imshow(abs(img_3),cmap ='plasma',alpha = 0.7)

<matplotlib.image.AxesImage at 0x2a9a3319c08>

Padding Lines of K-space to increase FOV

k_x = np.zeros((256,256),dtype='complex')
k_x[original] = k_first_x_original
img_1_pad = np.fft.fftshift(np.fft.fft2(k_x))

k_x_shift = np.zeros((256,256),dtype='complex')
k_x_shift[shifted] = k_first_x_shifted
img_2_pad = np.fft.fftshift(np.fft.fft2(k_x_shift))

k_y = np.zeros((256,256),dtype='complex')
k_y[:,original] = k_first_y_original
img_3_pad = np.fft.fftshift(np.fft.fft2(k_y))

k_y_shift = np.zeros((256,256),dtype='complex')
k_y_shift[:,shifted] = k_first_y_shifted
img_4_pad = np.fft.fftshift(np.fft.fft2(k_y_shift))

plt.figure(figsize=(30,20))


plt.subplot(321)
plt.imshow(abs(img_1_pad),cmap='PiYG')

plt.subplot(323)
plt.imshow(abs(img_2_pad),cmap ='PiYG')

plt.subplot(325)
plt.imshow(abs(img_2_pad),cmap ='PiYG')
plt.imshow(abs(img_1_pad),cmap ='PiYG',alpha = 0.7)

plt.subplot(322)
plt.imshow(abs(img_3_pad),cmap='plasma')

plt.subplot(324)
plt.imshow(abs(img_4_pad),cmap ='plasma')

plt.subplot(326)
plt.imshow(abs(img_4_pad),cmap ='plasma')
plt.imshow(abs(img_3_pad),cmap ='plasma',alpha = 0.7)

<matplotlib.image.AxesImage at 0x2a9a6c9de08>